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| {{about|a class of algebraic structures of the same signature|the set of solutions to a system of polynomial equations|Algebraic variety}}
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| In [[mathematics]], specifically [[universal algebra]], a '''variety of algebras''' is the [[class (set theory)|class]] of all [[algebraic structure]]s of a given [[signature (logic)|signature]] satisfying a given set of [[mathematical identity|identities]]. Equivalently, a variety is a class of algebraic structures of the same signature that is closed under the taking of [[homomorphism|homomorphic]] images, [[subalgebra]]s and [[Direct product|(direct) products]]. In the context of [[category theory]], a variety of algebras is usually called a '''finitary algebraic category'''.
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| A '''covariety''' is the [[class (set theory)|class]] of all [[F-coalgebra|coalgebraic structures]] of a given [[signature (logic)|signature]].
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| A variety of algebras should not be confused with an [[algebraic variety]]. Intuitively, a variety of algebras is an equationally defined '''collection of algebras''', while an algebraic variety is an equationally defined '''collection of elements from a single algebra'''. The two are named alike by analogy, but they are formally quite distinct and their theories have little in common. | |
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| ==Birkhoff's theorem==<!-- This section is linked from [[Universal algebra]] -->
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| [[Garrett Birkhoff]] proved equivalent the two definitions of variety given above, a result of fundamental importance to universal algebra and known as '''Birkhoff's theorem''' or as the '''HSP theorem'''. '''H''', '''S''', and '''P''' stand, respectively, for the [[closure (mathematics)|closure]] [[Operation (mathematics)|operations]] of [[homomorphism]], [[subalgebra]], and [[direct product|product]]''.
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| An equational class for some [[signature (logic)|signature]] Σ is the collection of all models, in the sense of [[model theory]], that satisfy some set ''E'' of ''equations'', asserting equality between terms. A model ''satisfies'' these equations if they are true in the model for any valuation of the variables. The equations in ''E'' are then said to be [[mathematical identity|identities]] of the model. Examples of such identities are the [[commutative law]], characterizing [[commutative algebra]]s, and the [[absorption law]], characterizing [[lattice (order)|lattice]]s.
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| It is simple to see that the class of algebras satisfying some set of equations will be closed under the HSP operations. Proving the [[Theorem#Converse|converse]] —classes of algebras closed under the HSP operations must be equational— is much harder.
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| ==Examples==
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| The class of all [[semigroup]]s forms a variety of algebras of signature (2). A sufficient defining equation is the associative law:
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| ::<math>x(yz) = (xy)z.</math>
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| It satisfies the HSP closure requirement, since any homomorphic image, any subset closed under multiplication and any direct product of semigroups is also a semigroup.
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| The class of [[group (mathematics)|groups]] forms a class of algebras of signature
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| (2,1,0), the three operations being respectively ''multiplication'', ''inversion'' and ''identity''.
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| Any subset of a group closed under multiplication, under inversion and under identity (i.e.
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| containing the identity) forms a subgroup. Likewise, the collection of groups is closed under homomorphic image and under direct product. Applying Birkhoff's theorem, this is sufficient to tell us that the groups form a variety, and so it should be defined by a collection of identities. In fact, the familiar axioms of associativity, inverse and identity form one suitable set of identities:
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| :<math>x(yz) = (xy)z</math>
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| :<math>1 x = x 1 = x</math>
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| :<math>x x^{-1} = x^{-1} x = 1.</math>
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| A '''subvariety''' of a variety ''V'' is a subclass of ''V'' that has the same signature as ''V'' and is itself a variety. Notice that although every group becomes a semigroup when the identity as a constant is omitted (and/or the inverse operation is omitted), the class of groups does ''not'' form a subvariety of the variety of semigroups because the signatures are different. On the other hand the class of [[abelian group]]s is a subvariety of the variety of groups because it consists of those groups satisfying <math>xy = yx,</math> with no change of signature. Viewing a variety V and its homomorphisms as a [[category theory|category]], a subclass U of V that is itself a variety is a subvariety of V implies that U is a full subcategory of V, meaning that for any objects a, b in U, the homomorphisms from a to b in U are exactly those from a to b in V. On the other hand there is a sense in which [[Boolean algebra]]s and [[Boolean ring]]s can be viewed as subvarieties of each other even though they have different signatures, because of the translation between them allowing every Boolean algebra to be understood as a Boolean ring and conversely; in this sort of situation the homomorphisms between corresponding structures are the same.
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| ==Pseudovariety of finite algebras==
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| Since varieties are closed under arbitrary cartesian products, all non-trivial varieties contain infinite algebras. It follows that the theory of varieties is of limited use{{citation needed|date=April 2012}} in the study of finite algebras, where one must often apply techniques particular to the finite case. Attempts have been made to develop a finitary analogue of the theory of varieties.
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| A '''pseudovariety''' is usually defined to be a class of algebras of a given signature, closed under the taking of homomorphic images, subalgebras and finitary direct products. Not every author assumes that all algebras on a pseudovariety are finite; if this is the case, one sometimes talks of a '''variety of finite algebras'''. For pseudovarieties, there is no general finitary counterpart to Birkhoff's theorem but in many cases the introduction of a more complex notion of equations allows similar results to be derived.<ref>E. g., B. Banaschewski, ''The Birkhoff Theorem for varieties of finite algebras
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| '', [[Algebra Universalis]], Volume 17, Number 1 (1983), 360-368, DOI: 10.1007/BF01194543</ref>
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| Pseudovarieties are of particular importance in the study of finite [[semigroup]]s and
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| hence in [[formal language|formal language theory]]. [[Eilenberg's theorem]], often referred to as the ''variety theorem'' describes a natural correspondence between varieties of [[regular language]]s and pseudovarieties of finite semigroups.
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| ==Category theory==
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| If ''A'' is a finitary algebraic category, then the [[forgetful functor]]
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| :<math>U:A\to\bold{Set}</math>
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| is [[monad (category theory)|monadic]]. Even more, it is ''strictly monadic'', in that the [[comparison functor]]
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| :<math>K:A\to \bold{Set}^{\mathbb{T}}</math>
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| is an isomorphism (and not just an equivalence).<ref>Saunders Mac Lane, ''Categories for the Working Mathematician'', Springer. ''(See p. 152)''</ref> Here, <math>\bold{Set}^{\mathbb{T}}</math> is the [[Eilenberg–Moore category]] on <math>\bold{Set}</math>. In general, one says a category is an '''algebraic category''' if it is monadic over <math>\bold{Set}</math>. This is a more general notion than "finitary algebraic category" (the notion of "variety" used in universal algebra) because it admits such categories as '''CABA''' (complete atomic Boolean algebras) and '''CSLat''' (complete semilattices) whose signatures include infinitary operations. In those two cases the signature is large, meaning that it forms not a set but a proper class, because its operations are of unbounded [[arity]]. The algebraic category of [[sigma algebra]]s also has infinitary operations, but their arity is countable whence its signature is small (forms a set).
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| == See also ==
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| * [[Quasivariety]]
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| ==Notes==
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| <references/>
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| ==References==
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| Two monographs available free online:
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| * Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. ''[http://www.thoralf.uwaterloo.ca/htdocs/ualg.html A Course in Universal Algebra.]'' Springer-Verlag. ISBN 3-540-90578-2.
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| * Jipsen, Peter, and Henry Rose, 1992. ''[http://www1.chapman.edu/~jipsen/JipsenRoseVoL.html Varieties of Lattices]'', Lecture Notes in Mathematics 1533. Springer Verlag. ISBN 0-387-56314-8.
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| [[Category:Universal algebra]]
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